Re: [Haskell-cafe] From monads to monoids in a small category

On Tue, Sep 4, 2012 at 4:21 PM, Alexander Solla wrote:

On Tue, Sep 4, 2012 at 3:39 AM, Alberto G. Corona wrote:

...

"Monads are monoids in the category of endofunctors"

This Monoid instance for the endofunctors of the set of all elements of (m a) typematch in Haskell with FlexibleInstances:

instance Monad m => Monoid (a -> m a) where mappend = (>=>) -- kleisly operator mempty = return

The objects of a Kliesli category for a monad m aren't endofunctors. You want something like:

instance Monad m => Monoid (m a -> m (m a)) where ...

/These/ are endofunctors, in virtue of join transforming an m (m a) into an (m a).

Actually, even these aren't endofunctors, for a similar reason that : you "really" want something like instance Monad m => Monoid (m a -> m a) where mempty = id mappend = undefined -- exercise left to the reader (i.e., you want to do plumbing through the Eilenberg-Moore category for a monad, instead of the Kliesli category for a monad -- my last message exposes the kind of plumping you want, but not the right types.)